# Estimation of Distribution Algorithm for Energy-Efficient Scheduling in Turning Processes

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Statement

#### 2.1. Motivating Example

#### 2.2. Problem Description

#### 2.3. Analysis of Turning Operations

_{ik}) can be defined as: job i is handled on lathe k, and it generally needs to pass five steps which are job-loading, lathe starting, job cutting, lathe stopping, and job-unloading. Because a turning operation of O

_{ik}contains five sub-operations, the total processing time is the summation of them. Therefore, the total processing time of O

_{ik}can be achieved with Equation (1).

#### 2.4. Analysis of Energy Consumption

^{F}is a coefficient related to materials and cutting conditions. K

^{F}is also a coefficient for cutting force. The cutting force can be obtained with Equation (6) since the value of these coefficients can be found in Machinery Handbook.

## 3. Modeling

**Indices**

i | Index of jobs, $i\in \{1,2\cdots ,n\}$ |

j | Index of stages, $j\in \{1,2\cdots ,S\}$ |

k | Index of machines, $k\in \{1,2,\cdots ,M\}$ |

t | Index of event points, $t\in \{1,2,\cdots ,n\}$ |

l | Index of spindle speed levels, $l\in \{1,2,\cdots ,\mathrm{L}\}$ |

**Parameters**

${K}_{j}$ | Set of machines in stage j |

${N}_{ik}$ | Set of spindle speed levels where job i can be processed on machine k |

Mv | Positive constant large enough |

${n}_{kl}^{c}$ | Spindle speed of machine k at level l |

${V}_{ik}^{z}$ | Total removed volume for O_{ik} |

$a{p}_{ik}$ | Depth of cutting for O_{ik} |

f_{ik} | Feed rate for O_{ik} |

${d}_{i}^{0}$ | Semi-product diameter for O_{ik} |

${C}_{ik}^{F}$ | Coefficients of cutting force for O_{ik} |

${K}_{ik}^{F}$ | Coefficients of cutting force for O_{ik} |

${E}_{kl}^{S}$ | Power consumption for starting machine for O_{ik} |

${E}_{kl}^{D}$ | Power consumption for stopping machine for O_{ik} |

${P}_{kl}^{0}$ | Power of machine k running without load at speed level l |

${T}_{kk\prime}^{d}$ | Transport time between machine k and machine k’ |

${b}_{k}^{m}$ | Additional load loss coefficient of machine k |

$a$ | Weight of energy consumption |

TCE_{0} | Normalizing parameter of energy consumption |

Cmax_{0} | Normalizing parameter of makespan |

**Binary Variables**

**Positive Variables**

${S}_{kt}$ | Start time of the event t at machine k |

${F}_{kt}$ | Finish time of the event t at machine k |

$C\mathrm{max}$ | Maximum completion time, i.e., makespan |

$TCE$ | Energy consumption |

#### 3.1. Mathematical Model

_{0}and Cmax

_{0}, are applied as two normalizing parameters, and they are obtained by a heuristic rule in this paper. Equations (13) and (14) both ensure each job is processed once at any stage. Equation (15) ensures that one of the available spindle speeds is selected when a job is assigned to a machine. Equation (16) controls that, at most, one job is processed in an event point. Equation (17) controls that one machine is available at an event point only after its previous jobs are completed. Equation (18) ensures the completion time of an event point is equal to the sum of the start time and processing time. Equation (19) limits that the starting time of each job in any stage is at least equal to the total time of the completion time in the previous stage and the transport time. Equation (20) controls that the completion time of any event point on a machine is at most equal to the start time of the subsequent event point on the same machine. Equation (21) ensures that TCE is the summation of energy consumption of all turning operations. Equation (22) controls that the Cmax is greater than or equal to the completion times of the last event point on all machines.

#### 3.2. Heuristic Rule for Normalizing Parameters

_{0}and Cmax

_{0}are obtained by the heuristic rule, which can be described as follows.

**Step 1:**The theoretical linear cutting velocity ${v}_{ik}^{*}$ is calculated with Equation (23), the theoretical spindle speed is obtained with Equation (8), and then the real level of spindle speed noted as l* is determined according to machine operating instructions.**Step 2:**The total processing time of one turning operation (${T}_{ikl*}^{Z}$) is calculated by Equation (2), the average processing time ($\overline{{T}_{i}}$) of job i in all stages is calculated with Equation (24), and then the scheduling scheme is obtained by the following three-step circulation.**Step 2.1:**Set j = 1 and sequence jobs by the ascending order of $\overline{{T}_{i}}$, denote the sequence as ${\mathsf{\pi}}_{t}$ and set ${T}_{{\mathsf{\pi}}_{1},1}^{c}=0$.**Step 2.2:**Assign the first free machine noted as $k*$ to process jobs in stage j, and calculate the completion time of jobs ${T}_{{\mathsf{\pi}}_{t},j}^{c}$ according to Equation (25). Then, calculate the consumed energy for cutting each job in stage j with Equation (11), and calculate the total energy consumption of stage j by ${C}_{j}^{z}={\displaystyle \sum _{k\in {K}_{j}}{\displaystyle \sum _{i}{C}_{ik{l}^{*}}^{z}}}$.Terminate this circulation when stage j is the last stage or go on to step 2.3.**Step 2.3:**Reorder jobs and update ${\mathsf{\pi}}_{t}$ in the ascending order of ${T}_{{\mathsf{\pi}}_{t},j}^{c}$, then set j = j + 1 and return to step 2.2.

**Step 3:**TEC_{0}can be obtained by $\sum _{j}{C}_{j}^{z}$, and $C{\mathrm{max}}_{0}$ can be determined by $\underset{i}{\mathrm{max}}\overline{{T}_{i}^{c}}$. Detailed explanations are described with Equations (23)–(25)$${v}_{ik}^{*}=\frac{{C}_{v}}{{T}^{{z}_{v}}a{p}_{ik}^{{x}_{v}}{f}_{ik}^{{y}_{v}}}$$$${\overline{T}}_{i}={\displaystyle \sum _{j}({\displaystyle \sum _{k\in {K}_{j}}{T}_{ikl*}^{z}}/{m}_{j}})/S$$$${T}_{{\mathsf{\pi}}_{t},j}^{c}=\{\begin{array}{l}\mathrm{max}\{{T}_{{\mathsf{\pi}}_{t},j-1}^{c},{T}_{{\mathsf{\pi}}_{t-1},j}^{c}\}+{T}_{{\mathsf{\pi}}_{t},k*l*}^{z},\forall t>1,j>1,\\ \mathrm{max}\{{T}_{{\mathsf{\pi}}_{t},j-1}^{c},0\}+{T}_{{\mathsf{\pi}}_{t},k*l*}^{z},\begin{array}{c}\forall t=1,j>1\end{array}\\ \mathrm{max}\{0,{T}_{{\mathsf{\pi}}_{t-1},j}^{c}\}+{T}_{{\mathsf{\pi}}_{t},k*l*}^{z},\begin{array}{c}\forall t>1,j=1\end{array}\end{array}$$

_{j}is the number of parallel machines in stage j, and K

_{j}is the set of machines.

## 4. EDA Algorithm

#### 4.1. Encoding, Decoding, and Dominant Individuals

_{size}is the population size and n the number of jobs.

**Step 1:**Choose an individual from the population, obtain its job sequence in the first stage, and set t = 1, $TE{C}^{*}=0$, ${T}_{{\mathsf{\pi}}_{1},1}^{c}=0$ and ${T}_{{\mathsf{\pi}}_{t},k}^{m}=0$.**Step 2:**Determine the processing machine and the spindle speed of all the jobs in current stage.**Step 2.1:**Calculate the processing time (${T}_{{\mathsf{\pi}}_{t}kl}^{z}$) and energy consumption (${E}_{{\mathsf{\pi}}_{t},k,l}$) of the current job (${\mathsf{\pi}}_{t}$) at all alternative speed levels on all available machines by Equations (2) and (11), respectively.**Step 2.2:**Calculate the completion time (${T}_{{\mathsf{\pi}}_{t},k,l}^{o}$) of the job at all alternative speed levels on all available machines by Equation (26). Set $C{\mathrm{max}}_{{\mathsf{\pi}}_{t},k,l}={T}_{{\mathsf{\pi}}_{t},k,l}^{o}$ and $TE{C}_{{\mathsf{\pi}}_{t},k,l}=TE{C}^{*}+{E}_{{\mathsf{\pi}}_{t},k,l}$.$${T}_{{\mathsf{\pi}}_{t},k,l}^{o}=\{\begin{array}{l}\mathrm{max}\{{T}_{{\mathsf{\pi}}_{t,}j-1}^{c}+{T}_{{k}^{\$}k}^{d},{T}_{{\mathsf{\pi}}_{t-1},k}^{m}\}+{T}_{{\mathsf{\pi}}_{t}kl}^{z},\text{}\forall t1,j1,k\in {K}_{j},l\in {N}_{\mathsf{\pi}t,k}\\ \mathrm{max}\{{T}_{{\mathsf{\pi}}_{t},j-1}^{c}+{T}_{{k}^{\$}k}^{d},0\}+{T}_{{\mathsf{\pi}}_{t}kl}^{z},\text{\hspace{1em}\hspace{1em}}\forall t=1,j1,k\in {K}_{j},l\in {N}_{\mathsf{\pi}t,k}\\ \mathrm{max}\{0,{T}_{{\mathsf{\pi}}_{t-1}k}^{m}\}+{T}_{{\mathsf{\pi}}_{t}kl}^{z},\text{\hspace{1em}\hspace{1em}\hspace{1em}}\forall t1,j=1,k\in {K}_{j},l\in {N}_{\mathsf{\pi}t,k}\end{array}$$**Step 2.3:**Calculate the weighted target value using Equation (12), select the machine and the speed level with the smallest weighted target value for the job, and mark the index of the machine and corresponding speed level with ${k}^{*}$ and ${l}^{*}$.**Step 2.4:**Set ${T}_{{\mathsf{\pi}}_{t},k*}^{m}={T}_{{\mathsf{\pi}}_{t},j}^{c}={T}_{{\mathsf{\pi}}_{t},k*,l*}^{o}$, $TE{C}^{*}=TE{C}^{*}+{E}_{{\mathsf{\pi}}_{t},k*,l*}$. If $t=n$, go to Step 3. Otherwise, set t = t + 1 and return to Step 2.1.

**Step 3:**If $j=S$, the decoding process terminates. Otherwise, determine the job sequence in ascending order of the completion times, set j = j + 1, t = 1, ${T}_{{\mathsf{\pi}}_{t},k}^{m}=0$, and return to Step 2. Calculate the final weighted target values of this individual using Equation (12).

#### 4.2. Population Updating Based on Probability Model

**Step 1:**Set the indicator function $(I{S}_{ti}^{l}(0))$to zero, and set all elements in the probability matrix $({\mathrm{Pr}}_{ti}(0))$ to $1/n$.**Step 2:**At the gth generation, if job $i$ is on position $t$ of dominant individual $l$, set $(I{S}_{ti}^{l}(g))$ to 1. Repeat this process till all dominant individuals, all jobs and all positions have been iterated. Calculate the total value of job $i$ on position $t$ and then yield the probability $({\mathrm{Pr}}_{ti}(g+1))$ by using Equation (27).$${\mathrm{Pr}}_{ti}(g+1)=(1-{a}^{s})\times {\mathrm{Pr}}_{ti}(g)+{a}^{s}\times {\displaystyle \sum _{l\in Sp}I{S}_{ti}^{l}(g)}/|Sp|,\forall g<G,t,i$$**Step 3:**Update the population according to ${\mathrm{Pr}}_{ti}(g+1)$ by the roulette approach. Terminate the algorithm if termination criterion is met; otherwise, set g = g + 1 and return to Step 2.

## 5. Verification and Discussion

_{0}and TEC

_{0}were determined for the three size experiments utilizing the heuristic rule in Section 3.2. Their values under small-scale, medium-scale, and large-scale circumstances were (3420 s, 29.07 MJ), (18,652 s, 289.17 MJ), and (26,763 s, 1294.1 MJ) respectively.

#### 5.1. Parameter Calibration of EDA

_{size}, the ratio of dominant population noted as $\mathsf{\eta}\%$ which is equal to $\left|Sp\right|/\left|{P}_{size}\right|$, and learning rate noted as a

^{s}. In this research, each factor has three levels: Psize (30, 50, 80), $\mathsf{\eta}$% (10%, 20%, 30%), and a

^{s}(10%, 20%, 30%). We adopt an orthogonal experiment whose size is L9 (3

^{3}) to calibrate these parameters, and the stopping criterion is the elapsed time of 80 s. The numerical results are obtained through the heurisitc rule. Then TEC

_{0}and Cmax

_{0}are used as parameters to calculate weighted goals according to Equation (12), where $a=0.8$. Finally, the AOV of each experiment is obtained as shown in Table 1, where AOV is the average value of the weighted targets for 30 tests.

#### 5.2. Experimental Results of the Motivating Example

#### 5.3. Discussion

## 6. Conclusions

- (1)
- A mixed integer nonlinear programming model is established by optimizing the spindle speed and scheduling scheme simultaneously, and subsequently, small-scale and medium-scale problems are solved by the GAMS/Dicopt;
- (2)
- For large-scale problems, an effective EDA algorithm is proposed in which a dispatching rule is embedded in decoding to specify the job with the earliest completion time to be first processed, and the population is updated utilizing the probability of the dominant individuals;
- (3)
- The experiment results show that: (1) the proposed algorithm can reduce the energy consumption to a certain extent and shorten the makespan to a large degree; and (2) there is a positive correlation between the energy consumption and makespan.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

Level | Speed (rpm) | E^{s} (J) | E^{D} (J) | T^{s} (s) | T^{D} (s) | P^{0} (w) |
---|---|---|---|---|---|---|

1 | 31.5 | 16.9 | 8 | 0.02 | 0.01 | 855 |

2 | 45 | 47.3 | 43 | 0.05 | 0.05 | 950 |

3 | 63 | 212 | 203 | 0.2 | 0.19 | 1060 |

4 | 90 | 1017 | 419 | 1 | 0.4 | 1010 |

5 | 125 | 1569 | 453 | 1.73 | 0.5 | 910 |

6 | 180 | 2122 | 463 | 2.38 | 0.52 | 890 |

7 | 250 | 3179 | 527 | 3.49 | 0.58 | 910 |

8 | 355 | 4065 | 547 | 4.45 | 0.59 | 920 |

9 | 500 | 4800 | 576 | 4.92 | 0.61 | 975 |

10 | 710 | 6003 | 696 | 5.61 | 0.65 | 1070 |

11 | 1000 | 7811 | 813 | 6.9 | 0.72 | 1115 |

12 | 1400 | 9809 | 837 | 8.67 | 0.74 | 1130 |

^{m}) is determined by the vibration noise, and the values of lathes R1, R2, F1, and F2 are 0.1; those of lathes R3, R4, F3, and F4 are 0.13; and the rest are 0.15.

Time | F1 | F2 | F3 | F4 | F5 | F6 |
---|---|---|---|---|---|---|

R1 | 5 | 6 | 7 | 8 | 9 | 10 |

R2 | 6 | 5 | 6 | 7 | 8 | 9 |

R3 | 7 | 6 | 5 | 6 | 7 | 8 |

R4 | 8 | 7 | 6 | 5 | 6 | 7 |

R5 | 9 | 8 | 7 | 6 | 5 | 6 |

Type | Material | Number | D (mm) | Length (mm) | d^{0} (mm) | t^{l} (min) | t^{u} (min) |
---|---|---|---|---|---|---|---|

1 | Cr12MoV | 8 | 66 | 1550 | 72 | 0.65 | 0.43 |

2 | Cr12MoV | 8 | 76 | 1620 | 83 | 0.72 | 0.48 |

3 | Cr12MoV | 6 | 85 | 1750 | 92 | 1.05 | 0.70 |

4 | 4Cr5MoSiV1 | 6 | 140 | 1526 | 150 | 1.15 | 0.77 |

5 | 4Cr5MoSiV1 | 6 | 236 | 1758 | 248 | 1.42 | 0.95 |

6 | 4Cr5MoSiV1 | 6 | 246 | 1846 | 260 | 1.50 | 1.00 |

7 | GCr15 | 4 | 202 | 1550 | 213 | 1.32 | 0.88 |

8 | GCr15 | 4 | 336 | 1620 | 350 | 1.85 | 1.23 |

9 | 45^{#} steel | 2 | 425 | 1720 | 442 | 2.08 | 1.39 |

10 | 45^{#} steel | 2 | 550 | 1720 | 570 | 2.35 | 1.57 |

11 | 3Cr2W8V | 4 | 360 | 1660 | 375 | 1.94 | 1.29 |

12 | 40Cr | 4 | 430 | 1520 | 446 | 2.02 | 1.35 |

Type | ap (mm) | f (mm/r) | v* (m/min) | Optional Level |
---|---|---|---|---|

1 | 2.75 | 0.3 | 140.3 | 9,10 |

2 | 3.25 | 0.3 | 136.8 | 9 |

3 | 3.25 | 0.3 | 136.8 | 9 |

4 | 4.7 | 0.4 | 95.4 | 6,7 |

5 | 5.6 | 0.5 | 85.9 | 4,5 |

6 | 6.6 | 0.5 | 83.8 | 4,5 |

7 | 5.1 | 0.5 | 106.5 | 5,6 |

8 | 6.5 | 0.6 | 96.4 | 4 |

9 | 7.9 | 0.7 | 88.7 | 3 |

10 | 9.4 | 0.8 | 81.9 | 2 |

11 | 7 | 0.6 | 95.3 | 3,4 |

12 | 7.5 | 0.6 | 94.3 | 3 |

Type | ap (mm) | f (mm/r) | v* (m/min) | Optional Level |
---|---|---|---|---|

1 | 0.25 | 0.1 | 250.4 | 11,12 |

2 | 0.25 | 0.1 | 250.4 | 11 |

3 | 0.25 | 0.1 | 250.4 | 11 |

4 | 0.30 | 0.15 | 224.6 | 9 |

5 | 0.40 | 0.15 | 215.1 | 7,8 |

6 | 0.40 | 0.15 | 215.1 | 7,8 |

7 | 0.40 | 0.15 | 215.1 | 8 |

8 | 0.50 | 0.2 | 196.4 | 6 |

9 | 0.60 | 0.2 | 191.1 | 5,6 |

10 | 0.60 | 0.25 | 182.8 | 4,5 |

11 | 0.50 | 0.2 | 196.4 | 6 |

12 | 0.50 | 0.2 | 196.4 | 5,6 |

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Combination | Level | AOV | ||
---|---|---|---|---|

Psize | η% | a^{s} | ||

1 | 80 | 30 | 0.1 | 0.9239 |

2 | 30 | 20 | 0.3 | 0.9237 |

3 | 80 | 10 | 0.3 | 0.9229 |

4 | 30 | 30 | 0.2 | 0.9243 |

5 | 50 | 30 | 0.3 | 0.9239 |

6 | 80 | 20 | 0.2 | 0.9239 |

7 | 50 | 20 | 0.1 | 0.9233 |

8 | 50 | 10 | 0.2 | 0.9230 |

9 | 30 | 10 | 0.1 | 0.9232 |

Solving Methods | Small Scale | Medium Scale | Large Scale | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Cmax (h) | TEC (MJ) | Z (%) | Time (s) | Cmax (h) | TEC (MJ) | Z (%) | Time (s) | Cmax (h) | TEC (MJ) | Z (%) | Time (s) | |

MINLP | 0.91 | 23.40 | 83.54 | 0.66 | 4.22 | 275.33 | 92.47 | 4.2 × 10^{3} | out of memory | >7.2 × 10^{4} | ||

EDA | 0.92 | 23.40 | 83.71 | 10.00 | 4.05 | 280.08 | 93.13 | 80.00 | 5.56 | 1223.2 | 90.57 | 180.00 |

Weights | TEC (MJ) | Cmax (h) |
---|---|---|

0.8 | 1223.2 | 5.56 |

0.6 | 1228.5 | 5.54 |

0.5 | 1234.0 | 5.50 |

0.4 | 1240.6 | 5.45 |

0.2 | 1243.1 | 5.43 |

0 | 1248.5 | 5.41 |

Relative difference | 2.03% | 2.76% |

Opt Goals | Opt Schemes | Weighted Objectives | Cmax | TEC | |||
---|---|---|---|---|---|---|---|

Mean (%) | Std. Deviation | Mean (s) | Std. Deviation | Mean (KJ) | Std. Deviation | ||

Obj_C | Opt_S | 0.9182 | 0.0009 | 20,138.03 | 114.48 | 1,088,720.23 | 595.05 |

Opt_o | 0.9766 | 0.0009 | 24,327.10 | 115.38 | 1,127,157.80 | 808.95 | |

Total | 0.9474 | 0.0293 | 22,232.57 | 2115.28 | 1,107,939.02 | 19,393.75 | |

Obj_e | Opt_S | 1.5976 | 0.0043 | 112,651.53 | 570.03 | 1,071,912.00 | 0.00 |

Opt_o | 1.7903 | 0.0036 | 134,722.73 | 475.01 | 1,111,200.00 | 0.00 | |

Total | 1.6940 | 0.0972 | 123,687.13 | 11,140.88 | 1,091,556.00 | 19,809.78 | |

Obj_t | Opt_S | 0.9212 | 0.0015 | 20,388.53 | 172.42 | 1,090,310.33 | 617.99 |

Opt_o | 0.9794 | 0.0020 | 24,503.33 | 236.15 | 1,129,263.87 | 1082.78 | |

Total | 0.9503 | 0.0294 | 22,445.93 | 2084.87 | 1,109,787.10 | 19,660.57 |

Effect | Value | F | Hypothesis df | Error df | Sig. | Partial Eta Squared | |
---|---|---|---|---|---|---|---|

Objs | Pillai’s Trace | 1.000 | 1,050,284.622 | 2.000 | 57.000 | 0.000 | 1.000 |

Wilks’ Lambda | 0.000 | 1,050,284.622 | 2.000 | 57.000 | 0.000 | 1.000 | |

Hotelling’s Trace | 36,852.092 | 1,050,284.622 | 2.000 | 57.000 | 0.000 | 1.000 | |

Roy’s Largest Root | 36,852.092 | 1,050,284.622 | 2.000 | 57.000 | 0.000 | 1.000 | |

Objs * methods | Pillai’s Trace | 0.997 | 8518.901 | 2.000 | 57.000 | 0.000 | 0.997 |

Wilks’ Lambda | 0.003 | 8518.901 | 2.000 | 57.000 | 0.000 | 0.997 | |

Hotelling’s Trace | 298.909 | 8518.901 | 2.000 | 57.000 | 0.000 | 0.997 | |

Roy’s Largest Root | 298.909 | 8518.901 | 2.000 | 57.000 | 0.000 | 0.997 |

Source | Type III Sum of Squares | df | Mean Square | F | Sig. | Partial Eta Squared |
---|---|---|---|---|---|---|

Intercept | 257.993 | 1 | 257.993 | 36,964,741.896 | 0.000 | 1.000 |

Methods | 0.478 | 1 | 0.478 | 68,527.167 | 0.000 | 0.999 |

Error | 0.000 | 58 | 6.979 × 10^{−6} |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, F.; Rao, Y.; Zhang, C.; Tang, Q.; Zhang, L. Estimation of Distribution Algorithm for Energy-Efficient Scheduling in Turning Processes. *Sustainability* **2016**, *8*, 762.
https://doi.org/10.3390/su8080762

**AMA Style**

Wang F, Rao Y, Zhang C, Tang Q, Zhang L. Estimation of Distribution Algorithm for Energy-Efficient Scheduling in Turning Processes. *Sustainability*. 2016; 8(8):762.
https://doi.org/10.3390/su8080762

**Chicago/Turabian Style**

Wang, Fang, Yunqing Rao, Chaoyong Zhang, Qiuhua Tang, and Liping Zhang. 2016. "Estimation of Distribution Algorithm for Energy-Efficient Scheduling in Turning Processes" *Sustainability* 8, no. 8: 762.
https://doi.org/10.3390/su8080762